Another problem associated with modeling lensing galaxies is
illustrated by the system HE0230-2130. There is not one lensing
galaxy, but two, as seen in figure 3. The second,
fainter galaxy provides the quadrupole moment that makes this a
4-image system and not a 2-image system. When we see two galaxies
separated by only a few kpc, we must wonder how the non-baryonic
matter is distributed. Are the halos for the two galaxies distinct?
Or have they merged? How many components should we model? And where
should their centers be? In such a case something with elements of
the [Williams & Saha (2000)]
form-free approach might be preferable
to a straightforward parameterized model.

Figure 3. The quadruple system HE0230-2130,
observed with the Baade 6.5-m telescope
through g' (left) and i' (right) filters. The two red objects
are lensing galaxies. The image between the two galaxies is a
saddlepoint of the time delay function.

The system with the best time delays is CLASS1608+656.
[Fassnacht, Xanthopoulos, Koopmans, & Rusin (2002)]
have measured four beautiful radio
lightcurves. All four time series faithfully reproduce the many bumps
and wiggles. The multiple delays in this case are good enough to
provide meaningful additional constraints to the deflections and
distortions. But modeling the system is not straightforward. As
with HE0230-2130, there
are two lensing galaxies (figure 4) that appear
to be interacting. A dust lane encircles them both. Again the question of
how the dark matter might be distributed looms as crucial.
[Koopmans et al. (2003)]
have worked exceedingly hard to constrain
this system, measuring positions for the images, the shape of the
ring, and measuring the velocity dispersion of the more massive lens.
We can nonetheless imagine a devil's advocate coming up with a
plausible dark matter distribution uncorrelated with the observed
galaxies that gives a very different value for the Hubble constant.

Figure 4. The quadruple system
CLASS1608+656, observed with the Hubble Space Telescope
through the F814W filter. The lensing galaxies are marked
G1 and G2. A dust lane can be seen encircling them.

The problem of multiple lenses can be serious even when one of the
galaxies is very much smaller than the principal lens. Consider the case
of RXJ0911+0554, shown in
figure 5, where the primary
lensing galaxy has a faint companion. Given its faintness and our
aversion to adding additional parameters, we might be tempted to
ignore it, as did
[Schechter (2000)]
in the model used by
Courbin to construct figure 1.
But allowing for a
mass at the position of this dwarf companion changes the predicted
time delay by 10%. Though the smaller galaxy is a factor of 10
fainter than the primary lensing galaxy, its effect is to move the
center of mass closer to the midpoint of the images, decreasing the
differences in path length.

Figure 5. The quadruple system
RXJ0911+0554, observed by the CASTLES consortium
with the Hubble Space
Telescope through the F160W filter. The primary lensing
galaxy G1
has a dwarf satellite G2. Allowing for the mass of the dwarf
changes the derived value of H0 by 10%.

There is an irony here in that the multiple lenses work to our
advantage in producing (or adding to) the quadrupole moments that give
us 4 images rather than 2. But at the same time they make modeling
considerably more difficult.